We show that, for several variants of the problem of compacting a grid
drawing of a graph to use the minimum number of rows or minimum area,
no good approximation algorithm is possible. We also develop
fixed-parameter tractable algorithms and approximation algorithms
showing that some of our inapproximability bounds are tight.
See the journal
version, "Inapproximability of
orthogonal compaction",
for some improvements and corrections.

The Bellman–Ford algorithm for single-source shortest paths in graphs
that may have negatively weighted edges but no negative cycles can be
sped up by a technique of Yen in which the graph is partitioned into two
directed acyclic subgraphs and edge relaxations alternate between these
two subgraphs. We show that choosing this partition randomly gains an
additional factor of 2/3 in running time.

We extend force-directed methods of graph drawing by adding a force that
pulls vertices towards the center of the drawing, with a strength
proportional to the centrality of the vertex. Gradually scaling up this
force helps avoid the tangling that would otherwise result from its use.

We study relational event data in which a collection of actors in a
social network have a sequence of pairwise interactions. Contiguous
subsequences of these interactions form graphs, and we develop efficient
data structures for querying the parameters of these graphs.

We show that testing whether a graph is 1-planar (drawable with at most
one crossing per edge) may be performed in polynomial and
fixed-parameter tractable time for graphs of bounded circuit rank,
vertex cover number, or tree-depth. However, it is NP-complete for
graphs of bounded treewidth, pathwidth, or bandwidth.

Many real-world graphs are k-almost-trees for small values of k: graphs
in which, in every biconnected component, removing a spanning tree
leaves at most k edges. We use kernelization methods to show that in
such graphs, the 1-page and 2-page crossing numbers can be computed quickly.

We construct small universal point sets for dominance drawings of
classes of acyclic graphs, by finding forbidden patterns in the
permutations determined by these drawings and proving the existence
of small superpatterns for the permutations with these patterns forbidden.
In particular, dominance drawings of the Hasse diagrams of width-2
partial orders have universal point sets of size
O(n3/2), derived from superpatterns of the same size
for the 321-avoiding permutations, and dominance drawings of st-planar
graphs have universal point sets of size
O(n log n), derived from superpatterns for
riffle shuffles.

We show that many standard graph drawing methods have algebraic
solutions described by polynomials that can have unsolvable Galois groups,
and that can have Galois groups whose order is divisible by large prime
numbers. As a consequence certain models of exact algebraic computation
are unable to construct these drawings.

We show how to express in monadic second-order logic the problems of
drawing a graph with a fixed
number of crossings on a one or two page book layout.
By applying Courcelle's theorem, we obtain fixed-parameter tractable
algorithms for these problems, parameterized by treewidth.

ERGMs (exponential random graph models) are used in social science to
describe probability distributions on graphs that are supposed to mimic
real-world social networks. However, we show that (with features that
are standard in the social science application) the distributions given
by these models can be computationally infeasible to sample from or
to approximate the probability of seeing a given graph.

We introduce the concept of a layered path decomposition, and show that
the layered pathwidth can be used to characterize the leveled planar
graphs. As a consequence we show that finding the minimum number of
tracks in a track layout of a given graph is NP-complete.
The GD version includes only the parts concerning track layout,
and uses the title "Track Layout is Hard".

We describe a system for transforming context-free grammars into
human-readable syntax diagrams, including optimizations that change the
structure of the grammar to make it more readable without affecting the
language described by the grammar.